On Some Matrix Theorems of Frobenius and McCoy

Abstract

McCoy, following Frobenius, studied a problem which can be described as follows. Let k be an arbitrary field, k^c its algebraic closure, and any algebra of n ⨯ n matrices over k which contains the identity I . Define a canonical ordering to be a set of n mappings λ of , or of a subset of , into k^c such that the sequence λ₁(A),λ₂(A), …, λ_n(A), for each A ∈ , consists of the characteristic values (roots of det(A — xI) = 0) of A, each with the right multiplicity. Define a canonical ordering to be a Frobenius ordering if, for all non-commutative polynomials f(x₁, x₂, … , x_m) and all finite subsets A₁, A₂, …, A_m of ,